#7728237

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#47456	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10,14,15)(11,16,18,13)(12,20,21,17)(19,22,24,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,22)(20,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(10,15)(11,18)(12,21)(22,23)(26,28)(31,33)(32,36)(38,39);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,12,14,21)(10,17,15,20)(11,19,18,24)(13,22,16,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,19,20,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7635433);